To solve the equation [tex]\(\sec(2\theta) = 3\)[/tex], we’ll proceed step-by-step using trigonometric identities to find the value of [tex]\(\cos(\theta)\)[/tex].
1. Express [tex]\(\sec(2\theta)\)[/tex] in terms of [tex]\(\cos(2\theta)\)[/tex]:
Recall that [tex]\(\sec(x) = \frac{1}{\cos(x)}\)[/tex]. Therefore, we can rewrite the given equation:
[tex]\[
\sec(2\theta) = 3 \implies \frac{1}{\cos(2\theta)} = 3
\][/tex]
2. Solve for [tex]\(\cos(2\theta)\)[/tex]:
Invert both sides of the equation to solve for [tex]\(\cos(2\theta)\)[/tex]:
[tex]\[
\cos(2\theta) = \frac{1}{3}
\][/tex]
3. Use the double-angle identity for cosine:
The double-angle identity for cosine is [tex]\(\cos(2\theta) = 2\cos^2(\theta) - 1\)[/tex]. Substitute [tex]\(\frac{1}{3}\)[/tex] for [tex]\(\cos(2\theta)\)[/tex]:
[tex]\[
\frac{1}{3} = 2\cos^2(\theta) - 1
\][/tex]
4. Solve the equation for [tex]\(\cos^2(\theta)\)[/tex]:
Isolate [tex]\(\cos^2(\theta)\)[/tex] by first adding 1 to both sides:
[tex]\[
\frac{1}{3} + 1 = 2\cos^2(\theta)
\][/tex]
Combine the fractions:
[tex]\[
\frac{1}{3} + \frac{3}{3} = \frac{4}{3}
\][/tex]
So, we have:
[tex]\[
\frac{4}{3} = 2\cos^2(\theta)
\][/tex]
5. Isolate [tex]\(\cos^2(\theta)\)[/tex]:
Divide both sides by 2:
[tex]\[
\cos^2(\theta) = \frac{\frac{4}{3}}{2} = \frac{4}{3} \times \frac{1}{2} = \frac{2}{3}
\][/tex]
6. Solve for [tex]\(\cos(\theta)\)[/tex]:
Take the square root of both sides to find [tex]\(\cos(\theta)\)[/tex]. Remember to include both the positive and negative roots:
[tex]\[
\cos(\theta) = \pm\sqrt{\frac{2}{3}}
\][/tex]
7. Simplify the square root:
[tex]\[
\cos(\theta) = \pm \sqrt{\frac{2}{3}} = \pm \frac{\sqrt{2}}{\sqrt{3}} = \pm \frac{\sqrt{2}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \pm \frac{\sqrt{6}}{3}
\][/tex]
Therefore, the solutions are:
- [tex]\(\cos(2\theta) = \frac{1}{3} \)[/tex]
- [tex]\(\cos(\theta) = \sqrt{\frac{2}{3}} \approx 0.816 \)[/tex]
- [tex]\(\cos(\theta) = -\sqrt{\frac{2}{3}} \approx -0.816 \)[/tex]
So the final values are:
- [tex]\(\cos(2\theta) = 0.333\)[/tex]
- [tex]\(\cos(\theta) \approx 0.816\)[/tex]
- [tex]\(\cos(\theta) \approx -0.816\)[/tex]
